Design: Blocked RCT, with 2 levels, and randomization done at level 1 (individual level).

Models: Constant treatment effects, fixed treatment effects, and random treatment effects.

Default parameters:

  • M = 3
  • J = 30
  • rho: \(\rho\) = 0.5
  • ATE ES = 0.125, 0.125, 0.125
  • R2.1: \(R^2_1\) = 0.1, 0.1, 0.1

Parameters by model type:

  • Omega: \(\omega_2 = 0\) for constant effects, \(\omega_2\) = 0.1 for fixed and random
  • ICC: \(\text{ICC}_2 = 0\) for constant and fixed effects, \(\text{ICC}_2\) = 0.2, 0.2, 0.2 for random effects

Assumptions:

  • Two-level design: \(\text{ICC}_3 = 0\), \(\omega_3 = 0\), \(K = 1\)

Power Validation

Base case

Varying school size

\(\bar{n}\) = 100

\(\bar{n}\) = 75

Varying R2

\(R^2_1 =\) 0.6, 0.6, 0.6

\(R^2_2 =\) 0.6, 0.6, 0.6

\(R^2_1 =\) 0, 0, 0 \(R^2_2 =\) 0, 0, 0

Varying rho

\(\rho =\) 0.2

\(\rho =\) 0.8

Varying true positives

ATE (ES) = 0.125, 0, 0

Varying ICC

\(\text{ICC}_2\) = 0.7, 0.7, 0.7

\(\text{ICC}_2\) = 0, 0, 0

Varying Omega

\(\omega_2\) = 0.8

\(\omega_2\) = 0

Kappa

\(\kappa\) = 0.4

MDES validation

## 
## 
## +------------+---------------+---------------+-------------+
## |    MTP     | Adjusted MDES | D1indiv Power | Target MDES |
## +============+===============+===============+=============+
## |    rawp    |     0.106     |     0.675     |    0.125    |
## +------------+---------------+---------------+-------------+
## | Bonferroni |     0.125     |     0.675     |    0.125    |
## +------------+---------------+---------------+-------------+
## |     BH     |     0.124     |     0.777     |    0.125    |
## +------------+---------------+---------------+-------------+
## |    Holm    |     0.125     |     0.754     |    0.125    |
## +------------+---------------+---------------+-------------+
## 
## Table: d2.1_m2fc
## 
## 
## +------------+---------------+---------------+-------------+
## |    MTP     | Adjusted MDES | D1indiv Power | Target MDES |
## +============+===============+===============+=============+
## |    rawp    |     0.105     |     0.669     |    0.125    |
## +------------+---------------+---------------+-------------+
## | Bonferroni |     0.124     |     0.669     |    0.125    |
## +------------+---------------+---------------+-------------+
## |     BH     |     0.125     |     0.781     |    0.125    |
## +------------+---------------+---------------+-------------+
## |    Holm    |     0.126     |     0.759     |    0.125    |
## +------------+---------------+---------------+-------------+
## 
## Table: d2.1_m2ff
## 
## 
## +------------+---------------+---------------+-------------+
## |    MTP     | Adjusted MDES | D1indiv Power | Target MDES |
## +============+===============+===============+=============+
## |    rawp    |      0.1      |     0.468     |    0.125    |
## +------------+---------------+---------------+-------------+
## | Bonferroni |     0.125     |     0.468     |    0.125    |
## +------------+---------------+---------------+-------------+
## |     BH     |     0.124     |     0.581     |    0.125    |
## +------------+---------------+---------------+-------------+
## |    Holm    |     0.124     |     0.54      |    0.125    |
## +------------+---------------+---------------+-------------+
## 
## Table: d2.1_m2fr

Sample size validation

## 
## 
## +------+-------------+-------------+---------------+
## | MTP  | Sample type | Sample size | D1indiv power |
## +======+=============+=============+===============+
## |  BH  |      J      |     30      |     0.767     |
## +------+-------------+-------------+---------------+
## | Holm |      J      |     30      |     0.749     |
## +------+-------------+-------------+---------------+
## 
## Table: d2.1_m2fc
## 
## 
## +------+-------------+-------------+---------------+
## | MTP  | Sample type | Sample size | D1indiv power |
## +======+=============+=============+===============+
## |  BH  |    nbar     |     49      |     0.772     |
## +------+-------------+-------------+---------------+
## | Holm |    nbar     |     51      |     0.754     |
## +------+-------------+-------------+---------------+
## 
## Table: d2.1_m2fc
## 
## 
## +------+-------------+-------------+---------------+
## | MTP  | Sample type | Sample size | D1indiv power |
## +======+=============+=============+===============+
## |  BH  |      J      |     30      |     0.777     |
## +------+-------------+-------------+---------------+
## | Holm |      J      |     31      |     0.759     |
## +------+-------------+-------------+---------------+
## 
## Table: d2.1_m2ff
## 
## 
## +------+-------------+-------------+---------------+
## | MTP  | Sample type | Sample size | D1indiv power |
## +======+=============+=============+===============+
## |  BH  |    nbar     |     50      |     0.778     |
## +------+-------------+-------------+---------------+
## | Holm |    nbar     |     51      |     0.754     |
## +------+-------------+-------------+---------------+
## 
## Table: d2.1_m2ff
## 
## 
## +------+-------------+-------------+---------------+
## | MTP  | Sample type | Sample size | D1indiv power |
## +======+=============+=============+===============+
## |  BH  |      J      |     30      |     0.578     |
## +------+-------------+-------------+---------------+
## | Holm |      J      |     31      |     0.544     |
## +------+-------------+-------------+---------------+
## 
## Table: d2.1_m2fr
## 
## 
## +------+-------------+-------------+---------------+
## | MTP  | Sample type | Sample size | D1indiv power |
## +======+=============+=============+===============+
## |  BH  |      J      |     30      |     0.578     |
## +------+-------------+-------------+---------------+
## | Holm |      J      |     31      |     0.544     |
## +------+-------------+-------------+---------------+
## 
## Table: d2.1_m2fr